12,611 research outputs found
Exact travelling wave solutions of non-linear reaction-convection-diffusion equations -- an Abel equation based approach
We consider quasi-stationary (travelling wave type) solutions to a general
nonlinear reaction-convection-diffusion equation with arbitrary, autonomous
coefficients. The second order nonlinear equation describing one dimensional
travelling waves can be reduced to a first kind first order Abel type
differential equation By using two integrability conditions for the Abel
equation (the Chiellini lemma and the Lemke transformation), several classes of
exact travelling wave solutions of the general reaction--convection-diffusion
equation are obtained, corresponding to different functional relations imposed
between the diffusion, convection and reaction functions. In particular, we
obtain travelling wave solutions for two non-linear second order partial
differential equations, representing generalizations of the standard diffusion
equation and of the classical Fisher--Kolmogorov equation, to which they reduce
for some limiting values of the model parameters. The models correspond to some
specific, power law type choices of the reaction and convection functions,
respectively. The travelling wave solutions of these two classes of
differential equations are investigated in detail by using both numerical and
semi-analytical methods.Comment: 35 pages, 3 figures, accepted for publication in JMP. arXiv admin
note: text overlap with arXiv:1409.060
Abel Dynamics of Titanium Dioxide Memristor Based on Nonlinear Ionic Drift Model
We give analytical solutions to the titanium dioxide memristor with arbitary
order of window functions, which assumes a nonlinear ionic drift model. As the
achieved solution, the characteristic curve of state is demonstrated to be a
useful tool for determining the operation point, waveform and saturation level.
By using this characterizing tool, it is revealed that the same input signal
can output completely different u-i orbital dynamics under different initial
conditions, which is the uniqueness of memristors. The approach can be regarded
as an analogy to using the characteristic curve for the BJT or MOS
transisitors. Based on this model, we further propose a class of analytically
solvable class of memristive systems that conform to Abel Differential
Equations. The equations of state (EOS) of the titanium dioxide memristor based
on both linear and nonlinear ionic drift models are typical integrable
examples, which can be categorized into this Abel memristor class. This large
family of Abel memristive systems offers a frame for obtaining and analyzing
the solutions in the closed form, which facilitate their characterization at a
more deterministic level.Comment: 5 pages, 3 figure
A class of exact solutions of the Li\'enard type ordinary non-linear differential equation
A class of exact solutions is obtained for the Li\'{e}nard type ordinary
non-linear differential equation. As a first step in our study the second order
Li\'{e}nard type equation is transformed into a second kind Abel type first
order differential equation. With the use of an exact integrability condition
for the Abel equation (the Chiellini lemma), the exact general solution of the
Abel equation can be obtained, thus leading to a class of exact solutions of
the Li\'{e}nard equation, expressed in a parametric form. We also extend the
Chiellini integrability condition to the case of the general Abel equation. As
an application of the integrability condition the exact solutions of some
particular Li\'{e}nard type equations, including a generalized van der Pol type
equation, are explicitly obtained.Comment: 18 pages, no figures; minor revisions, accepted for publication in
Journal of Engineering Mathematic
Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation
Given two polynomials we consider the following question: "how large
can the index of the first non-zero moment be,
assuming the sequence is not identically zero?". The answer to this
question is known as the moment Bautin index, and we provide the first general
upper bound: . The proof is
based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals
of certain algebraic functions.
The moment Bautin index plays an important role in the study of bifurcations
of periodic solution in the polynomial Abel equation
for polynomials and . In particular, our result
implies that for satisfying a well-known generic condition, the number of
periodic solutions near the zero solution does not exceed . This is the first such bound depending solely on the
degrees of the Abel equation
On Moment Condition and Center Condition for Abel Equation
In this paper we consider Abel equation , where and
are analytical functions. We proved that if the equation has a center at
, then the Moment Conditions, i. e.,
, is satisfied where
. Besides, we give partial a positive answer to a
conjecture proposed by Y. Lijun and T. Yun in 2001.Comment: arXiv admin note: text overlap with arXiv:1707.0266
Exact solutions of the Li\'enard and generalized Li\'enard type ordinary non-linear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator
We investigate the connection between the linear harmonic oscillator equation
and some classes of second order nonlinear ordinary differential equations of
Li\'enard and generalized Li\'enard type, which physically describe important
oscillator systems. By using a method inspired by quantum mechanics, and which
consist on the deformation of the phase space coordinates of the harmonic
oscillator, we generalize the equation of motion of the classical linear
harmonic oscillator to several classes of strongly non-linear differential
equations. The first integrals, and a number of exact solutions of the
corresponding equations are explicitly obtained. The devised method can be
further generalized to derive explicit general solutions of nonlinear second
order differential equations unrelated to the harmonic oscillator. Applications
of the obtained results for the study of the travelling wave solutions of the
reaction-convection-diffusion equations, and of the large amplitude free
vibrations of a uniform cantilever beam are also presented.Comment: 28 pages, no figures; minor modifications, accepted for publication
in Journal of Engineering Mathematic
Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates
In this paper, the exact analytical solution of the
Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric
form. By using the exact solution we investigate some explicit models
corresponding to fixed values of the parameters, and show that the numerical
solution reproduces exactly the analytical solution. We also show that the
generalization of the SIR model, including births and deaths, described by a
nonlinear system of differential equations, can be reduced to an Abel type
equation. The reduction of the complex SIR model with vital dynamics to an Abel
type equation can greatly simplify the analysis of its properties. The general
solution of the Abel equation is obtained by using a perturbative approach, in
a power series form, and it is shown that the general solution of the SIR model
with vital dynamics can be represented in an exact parametric form.Comment: 13 pages, 4 figures, accepted for publication in Applied Mathematics
and Computatio
A self-similar inhomogeneous dust cosmology
A detailed study of an inhomogeneous dust cosmology contained in a
-law family of perfect-fluid metrics recently presented by Mars and
Senovilla is performed. The metric is shown to be the most general orthogonally
transitive, Abelian, on solution admitting an additional homothety
such that the self-similar group is of Bianchi type VI and the fluid flow
is tangent to its orbits. The analogous cases with Bianchi types I, II, III, V,
VIII and IX are shown to be impossible thus making this metric privileged from
a mathematical viewpoint. The differential equations determining the metric are
partially integrated and the line-element is given up to a first order
differential equation of Abel type of first kind and two quadratures. The
solutions are qualitatively analyzed by investigating the corresponding
autonomous dynamical system. The spacetime is regular everywhere except for the
big bang and the metric is complete both into the future and in all spatial
directions. The energy-density is positive, bounded from above at any instant
of time and with an spatial profile (in the direction of inhomogeneity) which
is oscillating with a rapidly decreasing amplitude. The generic asymptotic
behaviour at spatial infinity is a homogeneous plane wave. Well-known dynamical
system results indicate that this metric is very likely to describe the
asymptotic behaviour in time of a much more general class of inhomogeneous
dust cosmologies.Comment: 17 pages, 3 postscript figures, uses psfig,sty, accepted for
publication in Class. Quantum Gra
Limit cycles of planar system defined by the sum of two quasi-homogeneous vecter fields
In this paper we consider the limit cycles of the planar system
where and
are quasi-homogeneous vector fields of degree and
respectively. We prove that under a new hypothesis, the maximal number of limit
cycles of the system is . We also show that our result can be applied to
some systems when the previous results are invalid. The proof is based on the
investigations for the Abel equation and the generalized-polar equation
associated with the system, respectively. Usually these two kinds of equations
need to be dealt with separately, and for both equations, an efficient approach
to estimate the number of periodic solutions is constructing suitable auxiliary
functions. In the present paper we develop a formula on the divergence, which
allows us to construct an auxiliary function of one equation with the auxiliary
function of the other equation, and vice versa
On the integrability of the Abel and of the extended Li\'{e}nard equations
We present some exact integrability cases of the extended Li\'{e}nard
equation , with and arbitrary constants,
while , , , and are arbitrary functions. The solutions
are obtained by transforming the equation Li\'{e}nard equation to an equivalent
first kind first order Abel type equation given by , with . As a first step in our study we obtain
three integrability cases of the extended quadratic-cubic Li\'{e}nard equation,
corresponding to and , by assuming that particular solutions of the
associated Abel equation are known. Under this assumption the general solutions
of the Abel and Li\'{e}nard equations with coefficients satisfying some
differential conditions can be obtained in an exact closed form. With the use
of the Chiellini integrability condition, we show that if a particular solution
of the Abel equation is known, the general solution of the extended quadratic
cubic Li\'{e}nard equation can be obtained by quadratures. The Chiellini
integrability condition is extended to generalized Abel equations with
and , and arbitrary and , thus allowing to
obtain the general solution of the corresponding Li\'{e}nard equation. The
application of the generalized Chiellini condition to the case of the reduced
Riccati equation is also considered.Comment: 13 pages, no figures, accepted for publication in Acta Mathematicae
Applicatae Sinica, English Serie
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